Let's solve both questions step by step.

First Question

(a) Find the length $x$ in mm

From the diagram:

• Two cylinders, each with a radius of 5 mm, are placed on top and bottom.
• The total vertical height, including both cylinders and the rectangular section, is 14 mm.
• The total height consists of:
  • The diameter of the top cylinder: $2 \times 5 = 10$ mm
  • The rectangular section: $4$ mm
  • The diameter of the bottom cylinder: $2 \times 5 = 10$ mm

So, the total height is: $10 + 4 + 10 = 14 \text{ mm}$

This confirms the given height is correct.

Since the length $x$ is just the diameter of one cylinder, we get: $x = 2 \times 5 = 10 \text{ mm}$

(b) Find the total surface area of the cylinder in mm² (in terms of $\pi$)

A cylinder's total surface area is given by: $\text{Total Surface Area} = 2\pi r h + 2\pi r^2$ where:

• $r = 5$ mm (radius)
• $h = x = 10$ mm (height)

Now, calculating each part:

• Curved surface area: $2\pi (5)(10) = 100\pi$
• Area of two circular faces: $2\pi (5)^2 = 50\pi$

Total surface area: $100\pi + 50\pi = 150\pi \text{ mm²}$

Second Question

Each cylinder has:

• Radius = $7$ cm
• Height = $3$ cm

Step 1: Calculate the surface area of one cylinder

$\text{Total Surface Area} = 2\pi r h + 2\pi r^2$

• Curved Surface Area: $2\pi (7)(3) = 42\pi$
• Area of two circular faces: $2\pi (7)^2 = 98\pi$

Total surface area: $42\pi + 98\pi = 140\pi$

Approximating $\pi \approx 3.14$: $140 \times 3.14 = 439.6 \text{ cm²}$

Step 2: Determine how many cylinders Valerie can paint

Total paint available: 3500 cm² $\frac{3500}{439.6} \approx 7.96$

Since Valerie can only paint complete cylinders, she can fully paint 7 cylinders.

Final Answers

1. First question:

   • $x = 10$ mm
   • Total surface area = $150\pi$ mm²

2. Second question:

   • Valerie can paint 7 cylinders completely.

Would you like a deeper explanation or another example?

Related Questions

1. How do you derive the formula for the surface area of a cylinder?
2. What is the difference between lateral surface area and total surface area?
3. How would the answer change if the radius of the cylinder doubled?
4. Can you generalize the number of cylinders Valerie can paint if she has $P$ cm² of paint?
5. What would be the volume of the given cylinders?

Tip:

For problems involving cylinders, always break the surface area into curved and circular parts to avoid mistakes.